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Berget and Rhoades asked whether the permutation representation obtained bythe action of $S_{n-1}$ on parking functions of length $n-1$ can be extended toa permutation action of $S_{n}$. We answer this question in the affirmative. Werealize our module in two different ways. The first description involves binaryLyndon words and the second involves the action of the symmetric group on thelattice points of the trimmed standard permutahedron.

Trimming the permutahedron to extend the parking space